Right now our SOLR setup is quite manual and we manually copy over some configuration files to the SOLR server. We also manually create the core. It is fairly simple to run SOLR since it's a Java application that can be extracted and run.
The README.md details how to set up the geoblacklight core and schema and our additions to it, which should always be checked into our repository after we make updates to the schema.
We haven't done formal backups of the SOLR data since it is fairly easy to repopulate from our database and in fact we have regenerated it on multiple occasions (every time we make schema changes or add facets).
The rsolr:reindex rake task is used to repopulate the SOLR index from the database and it runs quickly (somewhere in the range of a few minutes to a couple hours if I recall).
SOLR also offers a backup and restore functionality, so it could be manually backed up from one server and restored onto another. The SOLR backup and restore documentation gives information about how to back up a core and restore it.
While it's possible to use this option rather than repopulating the indexes from the database, I suspect it would not offer much of an advantage over running the repopulation rake task. There may be other reasons we want to keep backups or automate them, though.